Remark 13  If \(\tau _k = \rho \, \alpha _{k}^{\rm {BB2}}\)  (cf. (REF )), Lemma REF  does not hold for all the values of \(\rho \) : we must restrict ourselves to \(\rho > 2\) . Nevertheless, in the non-quadratic case, the gradient method is endowed with a line search, where bounds on the stepsize are provided by the user (see, e.g., [1]} and Section ). In this context, we may try also \(\rho \le 2\) , which corresponds to \(\xi _{\rm {low}}<\frac{1}{2}\) .

